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\title{About Planar Enneper}
\author{H. Karcher}
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   The surfaces Wavy Enneper,  Catenoid Enneper,  Planar Enneper,
and Double Enneper are finite total curvature minimal immersions
of the once or twice punctured sphere---shown with standard
polar coordinates. These surfaces illustrate how the different
types of ends can be combined in a simple way.

The pure Enneper surfaces ($aa = 0,\  dd = 0$ ) and the Planar
Enneper surfaces ($aa = 0, dd = 2$ ) with Weierstrass data:

Gauss map : $Gauss(z) = z^{ee+1}(1 + aa\, z^{ff})$   \hfill\break
Differential:  $ dh = scaling\cdot Gauss(z)/ z^{dd}\, dz$

have been re-discovered many times, because
the members of the associate family are \emph{congruent} surfaces
(as can be seen in the interesting associate family morphing!!) and
the Weierstrass integrals integrate to polynomial (respectively
rational) immersions. For $aa=0,\ dd=1$ one does not obtain a finite
total curvature surface, but a periodic surface that looks like a halfplane
with periodically attached Enneper pieces. One obtains larger pieces
of this {\it Wavy Plane} if one either increases $ee$ or the range of $v$.
The members of the associate family are also congruent, try
{\it Cyclic Associate Family Morph}.

For small $aa\ne 0$ one has wavy perturbations of the Enneper end.

  For the ($aa=0$)-examples ee is an integer valued parameter that
determines the degree of dihedral symmetry of the surfaces. 
Morphing the range of $u$ helps to
imagine these surfaces by starting with a plane minus a disk
and then observe how an Enneper end is attached. In the 
Set Morphing Dialog first press {\it Initiate to current parameters},
then choose $umax0=-0.5,\ umax1 = 0.6$.


 Formulas are taken from:

    H. Karcher, Construction of minimal surfaces, in ``Surveys in
     Geometry'', Univ. of Tokyo, 1989, and Lecture Notes No. 12,
     SFB 256, Bonn, 1989, pp. 1--96.


  For a discussion of techniques for creating minimal surfaces with
various qualitative features by appropriate choices of Weierstrass
data, see either [KWH], or pages 192--217 of [DHKW].

[KWH]  H. Karcher, F. Wei, and D. Hoffman, The genus one helicoid, and
         the minimal surfaces that led to its discovery, in ``Global Analysis
         in Modern Mathematics, A Symposium in Honor of Richard Palais'
         Sixtieth Birthday'', K. Uhlenbeck Editor, Publish or Perish Press, 1993

[DHKW] U. Dierkes, S. Hildebrand, A. Kuster, and O. Wohlrab,
           Minimal Surfaces I, Grundlehren der math. Wiss. v. 295
           Springer-Verlag, 1991



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